cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-36 of 36 results.

A250175 Numbers n such that Phi_15(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 11, 17, 23, 43, 46, 52, 53, 61, 62, 78, 84, 88, 89, 92, 99, 108, 123, 124, 141, 146, 154, 156, 158, 163, 170, 171, 182, 187, 202, 217, 219, 221, 229, 233, 238, 248, 249, 253, 264, 274, 275, 278, 283, 285, 287, 291, 296, 302, 309, 314, 315, 322, 325, 342, 346, 353, 356, 366, 368, 372, 377, 380, 384, 394, 404, 406, 411, 420, 425
Offset: 1

Views

Author

Eric Chen, Dec 24 2014

Keywords

Links

Crossrefs

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494(6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862(11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989(18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075(31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078(43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315(64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441(81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442(243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530(65536).

Programs

  • Mathematica
    Select[Range[600], PrimeQ[Cyclotomic[15, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)
  • PARI
    isok(n) = isprime(polcyclo(15, n)); \\ Michel Marcus, Jan 16 2015

Extensions

More terms from Vincenzo Librandi, Jan 16 2015

A250176 Numbers n such that Phi_20(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

4, 9, 11, 16, 19, 26, 34, 45, 54, 70, 86, 91, 96, 101, 105, 109, 110, 119, 120, 126, 129, 139, 141, 149, 171, 181, 190, 195, 215, 229, 260, 276, 299, 305, 309, 311, 314, 319, 334, 339, 369, 375, 414, 420, 425, 444, 470, 479, 485, 506, 519, 534, 540, 550
Offset: 1

Views

Author

Eric Chen, Dec 24 2014

Keywords

Links

Crossrefs

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494(6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862(11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989(18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075(31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078(43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315(64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441(81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442(243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530(65536).

Programs

  • Mathematica
    Select[Range[600], PrimeQ[Cyclotomic[20, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)
  • PARI
    isok(n) = isprime(polcyclo(20, n)); \\ Michel Marcus, Sep 29 2015

Extensions

More terms from Vincenzo Librandi, Jan 16 2015

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1
Offset: 0

Views

Author

Eric Chen, Apr 22 2015

Keywords

Comments

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

Examples

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Links

Crossrefs

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Programs

  • Mathematica
    Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]
  • PARI
    t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

Formula

T(m, n) = Phi_m(n)

A261436 Numbers k such that k^7-1 is a semiprime.

Original entry on oeis.org

3, 6, 14, 38, 60, 68, 72, 80, 128, 158, 180, 192, 264, 282, 294, 350, 450, 464, 548, 660, 710, 734, 798, 822, 878, 912, 942, 984, 998, 1052, 1188, 1194, 1224, 1280, 1284, 1382, 1424, 1482, 1494, 1512, 1550, 1554, 1572, 1608, 1622, 1668, 1700, 1710, 1790, 1802
Offset: 1

Views

Author

Vincenzo Librandi, Aug 21 2015

Keywords

Comments

Numbers k such that k-1 and k^6+k^5+k^4+k^3+k^2+k+1 are both prime.
Intersection of A008864 and A100330. - Michel Marcus, Aug 21 2015

Examples

			3 is in sequence because 3^7-1 = 2186 = 2*1093, where 2 and 1093 are both prime.

Links

Crossrefs

Cf. similar sequences listed in A261435.
Cf. A105041.
Cf. A008864, A100330.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..5000] | IsSemiprime(s) where s is n^7- 1];
  • Maple
    with(numtheory): A261436:=n->`if`(bigomega(n^7-1)=2, n, NULL): seq(A261436(n), n=1..2000); # Wesley Ivan Hurt, Aug 21 2015
select(n -> isprime(n-1) and isprime(n^6+n^5+n^4+n^3+n^2+n+1), [3,(2*i $i=2..10000)]); # Robert Israel, Aug 21 2015
  • Mathematica
    Select[Range[5000], PrimeOmega[#^7 - 1] == 2 &]
  • PARI
    isok(n)=bigomega(n^7-1)==2 \\ Anders Hellström, Aug 21 2015

A288939 Nonprime numbers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Original entry on oeis.org

1, 6, 14, 26, 38, 40, 46, 56, 60, 66, 68, 72, 80, 87, 93, 95, 115, 122, 126, 128, 146, 156, 158, 160, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 264, 266, 280, 282, 290, 294, 300, 303, 320, 341, 350, 363, 381, 395, 399, 404, 405, 417, 418, 436, 438, 447, 450
Offset: 1

Views

Author

Bernard Schott, Jun 19 2017

Keywords

Comments

A163268 Union {This sequence} = A100330.
The corresponding prime numbers k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 = 1111111_k are in A194194; all these Brazilian primes belong to A085104 and A285017.

Examples

			6 is in the sequence because 6^6 + 6^5 + 6^4 + 6^3 + 6^2 + 6 + 1 = 1111111_6 = 55987 which is prime.

Links

Crossrefs

Cf. A053716, A085104, A088550, A100330, A163268, A194194, A194257, A285017.

Programs

  • Maple
    for n from 1 to 200 do s(n):= 1+n+n^2+n^3+n^4+n^5+n^6;
if not isprime(n) and isprime(s(n)) then print(n,s(n)) else fi; od:
  • Mathematica
    Select[Range@ 450, And[! PrimeQ@ #, PrimeQ@ Total[#^Range[0, 6]]] &] (* Michael De Vlieger, Jun 19 2017 *)
  • PARI
    isok(n) = !isprime(n) && isprime(1+n+n^2+n^3+n^4+n^5+n^6); \\ Michel Marcus, Jun 19 2017
  • Python
    from sympy import isprime
A288939_list = [n for n in range(10**3) if not isprime(n) and isprime(n*(n*(n*(n*(n*(n + 1) + 1) + 1) + 1) + 1) + 1)] # Chai Wah Wu, Jul 13 2017

A258357 Numbers n such that cyclotomic polynomial value Phi(7,n!) is prime.

Original entry on oeis.org

0, 1, 2, 3, 13, 470, 2957
Offset: 1

Views

Author

Robert Price, May 27 2015

Keywords

Comments

Except for the values 0,1,2 and 3, terms correspond to probable primes.
a(8) > 6502.
Also, numbers n such that n! belongs to A100330. - Michel Marcus, May 30 2015

Examples

			3 is in the sequence because Phi(7,3!) = 1 + 6 + 6^2 + 6^3 + 6^4 + 6^5 + 6^6 = 55987 is prime.

Crossrefs

Cf. A002981, A002982, A046029, A200906.

Programs

  • Mathematica
    Select[Range[0, 6502], PrimeQ[Cyclotomic[7, #!]] &]
Previous Showing 31-36 of 36 results.

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